A combined volume penalization / selective frequency damping approach for immersed boundary methods: application to moving geometries (2306.00504v1)

Abstract: This work extends, to moving geometries, the immersed boundary method based on volume penalization and selective frequency damping approach [J. Kou, E. Ferrer, A combined volume penalization/selective frequency damping approach for immersed boundary methods applied to high-order schemes, Journal of Computational Physics (2023)]. To do so, the numerical solution inside the solid is decomposed into a predefined movement and an oscillatory part (spurious waves), where the latter is damped by an SFD approach combined with volume penalization. We challenge the method with two cases. First, a new manufactured solution problem is proposed to show that the method can recover high-order accuracy. Second, we validate the methodology by simulating the laminar flow past a moving cylinder, where improved accuracy of the combined method is reported.

• The paper introduces a new method combining VP and SFD to suppress spurious oscillations in immersed boundary methods for moving geometries.
• It validates the approach with manufactured solutions and flow past a moving cylinder, demonstrating high-order convergence and accurate dynamic behavior.
• The method enhances computational efficiency and accuracy in simulating fluid-structure interactions, paving the way for advanced CFD applications.

A Combined Volume Penalization and Selective Frequency Damping Approach for Immersed Boundary Methods

The paper authored by Jiaqing Kou and Esteban Ferrer presents an innovative method for improving the accuracy and efficiency of fluid dynamics simulations involving complex moving geometries. The work combines Volume Penalization (VP) with Selective Frequency Damping (SFD) in the context of Immersed Boundary Methods (IBM), an approach designed to handle fluid-structure interactions without the need for body-fitted meshes.

Background and Motivation

High-order methods like Flux Reconstruction (FR) and Discontinuous Galerkin (DG) have garnered attention for their superior accuracy and reduced numerical dissipation and dispersion errors. However, these methods face challenges with mesh generation, particularly around curved, body-fitted boundaries where high-order grids are required. To address this, the Immersed Boundary Method has been introduced, facilitating mesh generation by avoiding body-fitted meshes while still resolving complex flows using simple grids.

The authors' prior work established a robust IBM based on Volume Penalization for high-order discretizations. VP imposes boundary conditions by introducing penalizing source terms within solid regions, adding dissipation to mimic solid boundaries. However, when dealing with moving geometries, spurious oscillations can degrade accuracy. This motivated the integration with Selective Frequency Damping, which targets and suppresses these undesirable high-frequency oscillations.

Methodology

The proposed method reformulates SFD for moving boundaries, deriving new manufactured solutions for testing numerical convergence, and validating with Navier-Stokes equations. In this approach, the SFD operates by treating the numerical solution as comprising a base motion with an embedded oscillatory component. It strategically damps oscillations without affecting physical motions, relying on assumptions of spectral separation.

Several parameters govern the combined method's operation. The penalization parameter, $\eta$, is synchronized with the time step, serving as a primary tuning factor. The control parameter $\chi_f$, responsible for the damping strength, is set inversely proportional to $\eta$, while the filter width $\Delta$ is selected large enough to effectively filter out spurious frequencies.

Numerical Validation

Validation of the method entailed two scenarios: a manufactured solutions test and a dynamic case of flow past a moving circular cylinder. Both cases utilized a flux reconstruction scheme for spatial discretization.

1. Manufactured Solutions: By imposing a known analytical solution with designed spatiotemporal variation, the high-order convergence of the combined method was demonstrated. This test confirmed that the method maintains the expected convergence properties of the underlying high-order FR schemes.
2. Flow Past a Moving Cylinder: By simulating a cylinder undergoing sinusoidal motion in a fluid stream and observing vortex shedding phenomena, the method’s efficacy and accuracy were further established. The calculated frequency of lift-induced oscillations aligned with expected real-world values, showing the method's fidelity in capturing such intricate dynamical behaviors.

Implications and Future Research

The integration of VP and SFD in IBM for moving geometries presents a computationally efficient way to handle complex fluid-structure interaction problems with high accuracy. This approach can potentially be adapted to diverse applications involving dynamic boundary conditions and complex geometrical interactions.

Future developments may focus on optimizing parameter selection further and extending the applicability to turbulent flows and other sophisticated boundary configurations. Moreover, the framework may be leveraged for advanced studies in aerodynamics, biomechanics, and energy systems where fluid-structure interactions are critical.

In conclusion, this paper advances the field of computational fluid dynamics by presenting a novel method that significantly simplifies mesh generation complexities while addressing the accuracy challenges associated with moving geometries. Its application promises enhanced predictive capabilities for various scientific and engineering domains.